Challenging +1.8 This question requires recognizing that C represents a rotation matrix, computing CΒ² explicitly to form simultaneous equations, solving for a and b, then connecting a to cos(Ο/12) via the double angle relationship. While it involves multiple steps (matrix multiplication, solving equations, recognizing rotation matrices, and applying double angle formulas), each individual step is standard Further Maths technique. The main challenge is synthesizing these ideas and recognizing the geometric interpretation, making it moderately difficult but not requiring exceptional insight.
The matrix \(\mathbf{C} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}\), where \(a\) and \(b\) are positive real numbers,
and \(\mathbf{C}^2 = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\)
Use \(\mathbf{C}\) to show that \(\cos \frac{\pi}{12}\) can be written in the form \(\frac{\sqrt{m + n}}{2}\), where \(m\) and \(n\) are integers.
[7 marks]
Question 9:
9 | Explains that C2 represents
a rotation of or that C
ππ
represents a rotation of
6
ππ | 2.4 | E1 | 2 2
ππ βππ ππ βππ ππ βππ β2ππππ
οΏ½ οΏ½οΏ½ οΏ½ = οΏ½ 2 2οΏ½
ππ ππ ππ ππ 2ππππ ππ βππ
1
2ππππ = 2
2 2 β3
ππ βππ = 2
1
ππ = 4ππ
2 1 β3
2
ππ β16ππ = 2
4 2
16ππ β8β3ππ β1 = 0
2 β3+2
ππ = 4
since
οΏ½β3+2
ππ = 2 ππ > 0
C2 represents a rotation of , therefore C
ππ
represents a rotation of
6
1 ππ ππ
2οΏ½6οΏ½ = 12
So and
ππ ππ οΏ½β3+2
ππ = cos12 cos12 = 2
12
Squares matrix C with at
least two correct elements. | 3.1a | M1
Forms two simultaneous
equations in and using
their squared C and the
given C2 ππ ππ | 1.1a | M1
Forms two correct
simultaneous equations. | 1.1b | A1
Eliminates or and forms
a quadratic equation in
or ππ ππ 2
ππ | 1.1a | M1
2
Finππds the correct value of
2 β3+2 | 1.1b | A1
ππ = 4
Completes a rigorous
argument to show that
by
ππ οΏ½β3+2
explaining that C
cos12 = 2
represents a rotation of
ππ | 2.1 | R1
To12tal | 7
Q | Marking Instructions | AO | Marks | Typical Solution
The matrix $\mathbf{C} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, where $a$ and $b$ are positive real numbers,
and $\mathbf{C}^2 = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$
Use $\mathbf{C}$ to show that $\cos \frac{\pi}{12}$ can be written in the form $\frac{\sqrt{m + n}}{2}$, where $m$ and $n$ are integers.
[7 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2020 Q9 [7]}}